Efficient parallel and sequential algorithms for 4-coloring perfect planar graphs

We present an efficient algorithm for 4-coloring perfect planar graphs. The best previously known algorithm for this problem takesO(n ^3/2) sequential time, orO(log⁴ n) parallel time withO(n³) processors. The sequential implementation of our algorithm takesO(n logn) time. The parallel implementation of our algorithm takesO(log³ n) time withO(n) processors on a PRAM.

     

An Efficient Algorithm to Generate all Maximal Cliques on Trapezoid Graphs

In this paper, to find all maximal cliques of a trapezoid graph a set of intervals have been constructed by projecting the geometrical representation of the graph on the bottom line. The proposed algorithm for this purpose takes O(n ² + yn) time, where n is the number of vertices of the graph and y is the output size.     

基于整体退火遗传算法的低功耗最佳极性搜索

针对n变量逻辑函数在不同极性下所对应的XNOR/OR电路功耗和面积不同的特点,首先用信号概率传递算法和多输入XNOR/OR(同或/或)门的低功耗分解算法建立了XNOR/OR电路的功耗估计模型.在此基础上,将基于列表技术的极性转换算法和整体退火遗传算法相结合,提出了一种针对大规模XNOR/OR电路的低功耗最佳极性搜索算法.对8个较大规模MCNC Benchmark电路测试表明,该算法搜索到的最佳极性所对应的XNOR/OR电路与极性0时的XNOR/OR电路相比,平均节省功耗和面积分别达到了84.4%和65.2%.     

Finding least-weight subsequences with fewer processors

By restricting weight functions to satisfy the quadrangle inequality or the inverse quadrangle inequality, significant progress has been made in developing efficient sequential algorithms for the least-weight subsequence problem [10], [9], [12], [16]. However, not much is known on the improvement of the naive parallel algorithm for the problem, which is fast but demands too many processors (i.e., it takesO(log² n) time on a CREW PRAM with n³/logn processors). In this paper we show that if the weight function satisfies the inverse quadrangle inequality, the problem can be solved on a CREW PRAM in O(log² n log logn) time withn/log logn processors, or in O(log² n) time withn logn processors. Notice that the processor-time complexity of our algorithm is much closer to the almost linear-time complexity of the best-known sequential algorithm [12].     

基于MODCPSO算法的三值FPRM电路面积与延时优化

针对三值固定RM(fixed polarity reed-muller,FPRM)逻辑电路面积与延时综合优化问题进行了研究,提出了一种基于竞争行为多目标离散粒子群算法(Multi-Objective Discrete Competitive Particle Swarm Optimization,MODCPSO)的极性搜索方案。首先在MODCPSO算法中,引入竞争行为机制,将种群划分为不同的团队,从各个团队中随机抽取两个粒子进行比较,令较差的粒子向着较好的粒子进行速度和位置的更新。同时引入变异机制,令种群粒子能够跳出局部最优解,继续更新进化。然后,结合三值FPRM极性转换技术和MODCPSO算法搜索电路面积与延时的最佳极性。最后,利用PLA格式的MCNC Benchmark电路实现算法测试,并与DPSO算法、MODPSO算法进行了性能对比。实验结果验证了MODCPSO算法的有效性。     

Sull’ordinamento secondo un criterio di ottimalità decrescente delle politiche ammissibili,in un problema non lineare di ripartizione

In this Paper an algorithm is given, for a problem of allocation of resources, which allows to individnate directly the firstk policies (among those which invest on the whole the same quantity of resource as the optimal ones), policies being ordered suiving the rule of decreasing optimality. The algorithm is referred to the particular case of a discrete allocation problem, in which functions of the profit acquirable in productive processes, are given in a discrete set.

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Lavoro eseguito nell’ambito dell’attività del Gruppo di Ricerca Matematica n^o 38 (Università Ca’ Foscari, Venezia) del C.N.R. per l’anno accademico 1967–68.     

Construction of 1-resilient Boolean functions with optimum algebraic immunity

In this paper, for an integer n≥10, two classes of n-variable Boolean functions with optimum algebraic immunity (AI) are constructed, and their nonlinearities are also determined. Based on non-degenerate linear transforms to the proposed functions, we can obtain 1-resilient n-variable Boolean functions with optimum AI and high nonlinearity if n?1 is never equal to any power of 2.     

On the performance of Dijkstra’s third self-stabilizing algorithm for mutual exclusion and related algorithms

In Dijkstra (Commun ACM 17(11):643–644, 1974) introduced the notion of self-stabilizing algorithms and presented three such algorithms for the problem of mutual exclusion on a ring of n processors. The third algorithm is the most interesting of these three but is rather non intuitive. In Dijkstra (Distrib Comput 1:5–6, 1986) a proof of its correctness was presented, but the question of determining its worst case complexity—that is, providing an upper bound on the number of moves of this algorithm until it stabilizes—remained open. In this paper we solve this question and prove an upper bound of 3\frac1318 n² + O(n){3\frac{13}{18} n^2 + O(n)} for the complexity of this algorithm. We also show a lower bound of 1\frac56 n² - O(n){1\frac{5}{6} n^2 - O(n)} for the worst case complexity. For computing the upper bound, we use two techniques: potential functions and amortized analysis. We also present a new-three state self-stabilizing algorithm for mutual exclusion and show a tight bound of \frac56 n² + O(n){\frac{5}{6} n^2 + O(n)} for the worst case complexity of this algorithm. In Beauquier and Debas (Proceedings of the second workshop on self-stabilizing systems, pp 17.1–17.13, 1995) presented a similar three-state algorithm, with an upper bound of 5\frac34n²+O(n){5\frac{3}{4}n^2+O(n)} and a lower bound of \frac18n²-O(n){\frac{1}{8}n^2-O(n)} for its stabilization time. For this algorithm we prove an upper bound of 1\frac12n² + O(n){1\frac{1}{2}n^2 + O(n)} and show a lower bound of n ²−O(n). As far as the worst case performance is considered, the algorithm in Beauquier and Debas (Proceedings of the second workshop on self-stabilizing systems, pp 17.1–17.13, 1995) is better than the one in Dijkstra (Commun ACM 17(11):643–644, 1974) and our algorithm is better than both.     

Labeling Schemes for Dynamic Tree Networks

Distance labeling schemes are composed of a marker algorithm for labeling the vertices of a graph with short labels, coupled with a decoder algorithm allowing one to compute the distance between any two vertices directly from their labels (without using any additional information). As applications for distance labeling schemes concern mainly large and dynamically changing networks, it is of interest to study distributed dynamic labeling schemes. The current paper considers the problem on dynamic trees, and proposes efficient distributed schemes for it. The paper first presents a labeling scheme for distances in the dynamic tree model, with amortized message complexity O(log² n) per operation, where n is the size of the tree at the time the operation takes place. The protocol maintains O(log² n) bit labels. This label size is known to be optimal even in the static scenario. A more general labeling scheme is then introduced for the dynamic tree model, based on extending an existing static tree labeling scheme to the dynamic setting. The approach fits a number of natural tree functions, such as distance, separation level, and flow. The main resulting scheme incurs an overhead of an O(log n) multiplicative factor in both the label size and amortized message complexity in the case of dynamically growing trees (with no vertex deletions). If an upper bound on n is known in advance, this method yields a different tradeoff, with an O(log² n/log log n) multiplicative overhead on the label size but only an O(log n/log log n) overhead on the amortized message complexity. In the fully dynamic model the scheme also incurs an increased additive overhead in amortized communication, of O(log² n) messages per operation.     

Precision,Local Search and Unimodal Functions

We investigate the effects of precision on the efficiency of various local search algorithms on 1-D unimodal functions. We present a (1+1)-EA with adaptive step size which finds the optimum in O(log n) steps, where n is the number of points used. We then consider binary (base-2) and reflected Gray code representations with single bit mutations. The standard binary method does not guarantee locating the optimum, whereas using the reflected Gray code does so in Θ((log n)²) steps. A(1+1)-EA with a fixed mutation probability distribution is then presented which also runs in O((log n)²). Moreover, a recent result shows that this is optimal (up to some constant scaling factor), in that there exist unimodal functions for which a lower bound of Ω((log n)²) holds regardless of the choice of mutation distribution. For continuous multimodal functions, the algorithm also locates the global optimum in O((log n)²). Finally, we show that it is not possible for a black box algorithm to efficiently optimise unimodal functions for two or more dimensions (in terms of the precision used).     

Solving Connected Dominating Set Faster than 2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In the connected dominating set problem we are given an n-node undirected graph, and we are asked to find a minimum cardinality connected subset S of nodes such that each node not in S is adjacent to some node in S. This problem is also equivalent to finding a spanning tree with maximum number of leaves. Despite its relevance in applications, the best known exact algorithm for the problem is the trivial Ω(2 ⁿ ) algorithm that enumerates all the subsets of nodes. This is not the case for the general (unconnected) version of the problem, for which much faster algorithms are available. Such a difference is not surprising, since connectivity is a global property, and non-local problems are typically much harder to solve exactly. In this paper we break the 2 ⁿ barrier, by presenting a simple O(1.9407 ⁿ ) algorithm for the connected dominating set problem. The algorithm makes use of new domination rules, and its analysis is based on the Measure and Conquer technique. An extended abstract of this paper appeared in the proceedings of FSTTCS’06. Fedor V. Fomin was additionally supported by the Research Council of Norway.     

Novel optimized tree-based stack-type architecture for 2n-bit comparator at nanoscale with energy dissipation analysis

Comparator is an essential building block in many digital circuits such as biometric authentication, data sorting, and exponents comparison in floating-point architectures among others. Quantum-dot Cellular Automata (QCA) is a latest nanotechnology that overcomes the drawbacks of Complementary Metal Oxide Semiconductor (CMOS) technology. In this paper, novel area optimized 2n-bit comparator architecture is proposed. To achieve the objective, 1-bit stack-type and 4-bit tree-based stack-type (TB-ST) comparators are proposed using QCA. Then, two tree-based architectures of 4-bit comparators are arranged in two layers to optimize the number of quantum cells and area of an 8-bit comparator. Thus, this design can be extended to any 2n-bit comparator. Simulation results of 4-bit and 8-bit comparators using QCADesigner 2.0.3 show that there is a significant improvement in the number of quantum cells and area occupancy. The proposed TB-ST 8-bit comparator uses 2.5 clock cycles and 622 quantum cells with area occupancy of 0.49 µm² which is an improvement by 10.5% and 38%, respectively, compared to existing designs. Scaling it to a 32-bit comparator, the proposed architecture requires only 2675 quantum cells in an area of 2.05 µm² with a delay of 3.5 clock cycles, indicating 9.35% and 28.8% improvements, respectively, demonstrating the merit of the proposed architecture. Besides, energy dissipation analysis of the proposed TB-ST 8-bit comparator is simulated on QCADesigner-E tool, indicating average energy dissipation reduction of 17.3% compared to existing works.

     

A more efficient algorithm for the min-plus multiplication

The (min, + ) product C of two n × n matrices A and B is defined as C _ij = min_1≦k≦n A _ik + B _kj . This paper presents an algorithm to compute the (min, +) product of two n × n matrices. The algorithm follows the approach described by Fredman, but is faster than Fredman's own algorithm: its time complexity is either O(n ³/√log₂ n) or even O(n ^2.5√log₂ n), if one adheres to the uniform-cost RAM model faithfully.

This result implies the existence of O(n ³/√log₂ n) algorithms for the problems that (min, +) matrix multiplication is equivalent to, such as the all-pairs shortest paths problem.

As the presented algorithm uses operations on sets, the formal analysis of its time complexity raises a few interesting questions about the applicability of the standard RAM complexity model.     

Qualche contributo alla risoluzione delle disequazioni variazionali inR n

Riassunto Viene indicato un metodo di risoluzione di disequazioni variazionali in un convesso diR ⁿ definito da funzioni convesse non necessariamente affini: il metodo è la naturale estensione di quello indicato in [1] per il caso di convessi diR ⁿ definiti da funzioni affini.

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Here we show a method of computing solutions of variational inequalities in a convex subset ofR ⁿ defined by convex, not necessarily affine, functions: this method is the natural extension of that related in [1] for convex subset ofR ⁿ defined by affine functions.

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Lavoro estratto dalla Tesi di Laurea [5] svolta presso l'Istituto di Elaborazione della Informazione, Pisa, (Rel. A. Laratta).     

On the Sample Complexity of Weak Learning

In this paper, we study the sample complexity of weak learning. That is, we ask how many data must be collected from an unknown distribution in order to extract a small but significant advantage in prediction. We show that it is important to distinguish between those learning algorithms that output deterministic hypotheses and those that output randomized hypotheses. We prove that in the weak learning model, any algorithm using deterministic hypotheses to weakly learn a class of Vapnik-Chervonenkis dimension d(n) requires Ω ([formula]) examples. In contrast, when randomized hypotheses are allowed, we show that Θ (1) examples suffice in some cases. We then show that there exists an efficient algorithm using deterministic hypotheses that weakly learns against any distribution on a set of size d(n) with only O(d(n)^2/3) examples. Thus for the class of symmetric Boolean functions over n variables, where the strong learning sample complexity is Θ (n), the sample complexity for weak learning using deterministic hypotheses is Ω ([formula]) and O(n^2/3), and the sample complexity for weak learning using randomized hypotheses is Θ (1). Next we prove the existence of classes for which the distribution-free sample size required to obtain a slight advantage in prediction over random guessing is essentially equal to that required to obtain arbitrary accuracy. Finally, for a class of small circuits, namely all parity functions of subsets of n Boolean variables, we prove a weak learning sample complexity of Θ(n). This bound holds even if the weak learning algorithm is allowed to replace random sampling with membership queries, and the target distribution is uniform on {0, 1}ⁿ.     

虫孔路由Mesh上的连通分量算法及其应用

用倍增技术在带有Wormhole路由技术的n×n二维网孔机器上提出了时间复杂度为O(log²n)的连通分量和传递闭包并行算法,并在此基础上提出了一个时间复杂度为O(log³n)的最小生成树并行算法.这些都改进了Store-and-Forward路由技术下的时间复杂度下界O(n).同其他运行在非总线连接分布式存储并行计算机上的算法相比,此连通分量和传递闭包算法的时间复杂度是最优的.     

Efficient parallel algorithms for breadth first spanning forests of general graphs

Ghosh and Bhattacharjee propose [2] (Intern. J. Computer Math., 1984, Vol. 15, pp. 255-268) an algorithm of determining breadth first spanning trees for graphs, which requires that the input graphs contain some vertices, from which every other vertex in the input graph can be reached. These vertices are called starting vertices. The complexity of the GB algorithm is O(log² n) using O{n ³) processors. In this paper an algorithm, named BREADTH, also computing breadth first spanning trees, is proposed. The complexity is O(log² n) using O{n ³/logn) processors. Then an efficient parallel algorithm, named- BREADTHFOREST, is proposed, which generalizes algorithm BREADTH. The output of applying BREADTHFOREST to a general graph, which may not contain any starting vertices, is a breadth first spanning forest of the input graph. The complexity of BREADTHFOREST is the same as BREADTH.     

Perfect load balancing on the star interconnection network

In this paper, we use the regular distribution method to design a perfect load balancing algorithm for an n-star with a maximum error of 1 and a time complexity of 3n(n+1). This algorithm is based on the novel notion of leader trees. A second algorithm proposed in this paper as an enhancement to our first algorithm and uses an arbitrary spanning tree as the leader tree and has a worst time complexity of 2.25n ²−3n+0.75. We also discuss the issue of dynamically selecting the leader tree and hybrid load balancing algorithms in general. Furthermore, we present a hybrid algorithm for load balancing on the star interconnection network which benefits from a diffusion load balancing preprocessing phase and shows a smaller mean time complexity than our two first algorithms.     

A class of boolean functions with linear combinational complexity

In this paper we investigate the combinational complexity of Boolean functions satisfying a certain property, ⁿ_k,m. A function of n variables has the ⁿ_k,m property if there are at least m functions obtainable from each way of restricting it to a subset of n - - k variables. We show that the complexity of a ⁿ_3,5 function is no less than , and this bound cannot be much improved. Further, we find that for each k, there are ⁿ_k,2^k functions with complexity linear in n.     

Parallel algorithms for arrangements

We give the first efficient parallel algorithms for solving the arrangement problem. We give a deterministic algorithm for the CREW PRAM which runs in nearly optimal bounds ofO (logn log^* n) time andn ²/logn processors. We generalize this to obtain anO (logn log^* n)-time algorithm usingn ^d /logn processors for solving the problem ind dimensions. We also give a randomized algorithm for the EREW PRAM that constructs an arrangement ofn lines on-line, in which each insertion is done in optimalO (logn) time usingn/logn processors. Our algorithms develop new parallel data structures and new methods for traversing an arrangement.This work was supported by the National Science Foundation, under Grants CCR-8657562 and CCR-8858799, NSF/DARPA under Grant CCR-8907960, and Digital Equipment Corporation. A preliminary version of this paper appeared at the Second Annual ACM Symposium on Parallel Algorithms and Architectures [3].